Entrance laws at the origin of self-similar Markov processes in high dimensions

Kyprianou, Andreas; Rivero, Victor; Şengül, Batı; Yang, Ting

doi:10.1090/tran/8086

Entrance laws at the origin of self-similar Markov processes in high dimensions
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by Andreas E. Kyprianou, Victor Rivero, Batı Şengül and Ting Yang PDF

Trans. Amer. Math. Soc. 373 (2020), 6227-6299 Request permission

Abstract:

In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in $\mathbb {R}^d$, killed upon hitting the origin. Under suitable assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.

References

Larbi Alili, Loïc Chaumont, Piotr Graczyk, and Tomasz Żak, Inversion, duality and Doob $h$-transforms for self-similar Markov processes, Electron. J. Probab. 22 (2017), Paper No. 20, 18. MR 3622890, DOI 10.1214/17-EJP33
Gerold Alsmeyer, On the Markov renewal theorem, Stochastic Process. Appl. 50 (1994), no. 1, 37–56. MR 1262329, DOI 10.1016/0304-4149(94)90146-5
Gerold Alsmeyer and Fabian Buckmann, Fluctuation theory for Markov random walks, J. Theoret. Probab. 31 (2018), no. 4, 2266–2342. MR 3866614, DOI 10.1007/s10959-017-0778-9
Mrinal K. Ghosh, Aristotle Arapostathis, and Steven I. Marcus, Optimal control of switching diffusions with application to flexible manufacturing systems, SIAM J. Control Optim. 31 (1993), no. 5, 1183–1204. MR 1233999, DOI 10.1137/0331056
Søren Asmussen, Applied probability and queues, 2nd ed., Applications of Mathematics (New York), vol. 51, Springer-Verlag, New York, 2003. Stochastic Modelling and Applied Probability. MR 1978607
Søren Asmussen and Hansjörg Albrecher, Ruin probabilities, 2nd ed., Advanced Series on Statistical Science & Applied Probability, vol. 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. MR 2766220, DOI 10.1142/9789814282536
Rodrigo Bañuelos and Robert G. Smits, Brownian motion in cones, Probab. Theory Related Fields 108 (1997), no. 3, 299–319. MR 1465162, DOI 10.1007/s004400050111
Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564
Jean Bertoin, Homogeneous multitype fragmentations, In and out of equilibrium. 2, Progr. Probab., vol. 60, Birkhäuser, Basel, 2008, pp. 161–183. MR 2477381, DOI 10.1007/978-3-7643-8786-0_{8}
Jean Bertoin and Maria-Emilia Caballero, Entrance from $0+$ for increasing semi-stable Markov processes, Bernoulli 8 (2002), no. 2, 195–205. MR 1895890
Jean Bertoin and Mladen Savov, Some applications of duality for Lévy processes in a half-line, Bull. Lond. Math. Soc. 43 (2011), no. 1, 97–110. MR 2765554, DOI 10.1112/blms/bdq084
Jean Bertoin and Wendelin Werner, Stable windings, Ann. Probab. 24 (1996), no. 3, 1269–1279. MR 1411494, DOI 10.1214/aop/1065725181
Jean Bertoin and Marc Yor, The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes, Potential Anal. 17 (2002), no. 4, 389–400. MR 1918243, DOI 10.1023/A:1016377720516
Krzysztof Bogdan, Zbigniew Palmowski, and Longmin Wang, Yaglom limit for stable processes in cones, Electron. J. Probab. 23 (2018), Paper No. 11, 19. MR 3771748, DOI 10.1214/18-EJP133
M. E. Caballero and L. Chaumont, Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes, Ann. Probab. 34 (2006), no. 3, 1012–1034. MR 2243877, DOI 10.1214/009117905000000611
L. Chaumont and R. A. Doney, Corrections to: “On Lévy processes conditioned to stay positive” [Electron J. Probab. 10 (2005), no. 28, 948–961; MR2164035], Electron. J. Probab. 13 (2008), no. 1, 1–4. MR 2375597, DOI 10.1214/EJP.v13-466
Loïc Chaumont, Henry Pantí, and Víctor Rivero, The Lamperti representation of real-valued self-similar Markov processes, Bernoulli 19 (2013), no. 5B, 2494–2523. MR 3160562, DOI 10.3150/12-BEJ460
Kai Lai Chung and John B. Walsh, Markov processes, Brownian motion, and time symmetry, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer, New York, 2005. MR 2152573, DOI 10.1007/0-387-28696-9
Erhan Çinlar, Lévy systems of Markov additive processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 175–185. MR 370788, DOI 10.1007/BF00536006
Erhan Çinlar, Entrance-exit distributions for Markov additive processes, Math. Programming Stud. 5 (1976), 22–38. MR 445621, DOI 10.1007/bfb0120761
Robert C. Dalang and M.-O. Hongler, The right time to sell a stock whose price is driven by Markovian noise, Ann. Appl. Probab. 14 (2004), no. 4, 2176–2201. MR 2100388, DOI 10.1214/105051604000000747
Steffen Dereich, Leif Döring, and Andreas E. Kyprianou, Real self-similar processes started from the origin, Ann. Probab. 45 (2017), no. 3, 1952–2003. MR 3650419, DOI 10.1214/16-AOP1105
Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
Runhuan Feng and Yasutaka Shimizu, Potential measures for spectrally negative Markov additive processes with applications in ruin theory, Insurance Math. Econom. 59 (2014), 11–26. MR 3283205, DOI 10.1016/j.insmatheco.2014.08.001
P. J. Fitzsimmons and R. K. Getoor, Lévy systems and time changes, Séminaire de Probabilités XLII, Lecture Notes in Math., vol. 1979, Springer, Berlin, 2009, pp. 229–259. MR 2599212, DOI 10.1007/978-3-642-01763-6_{9}
P. J. Fitzsimmons, The quasi-sure ratio ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 3, 385–405 (English, with English and French summaries). MR 1625863, DOI 10.1016/S0246-0203(98)80016-2
Rodolphe Garbit, Brownian motion conditioned to stay in a cone, J. Math. Kyoto Univ. 49 (2009), no. 3, 573–592. MR 2583602, DOI 10.1215/kjm/1260975039
Rodolphe Garbit and Kilian Raschel, On the exit time from a cone for Brownian motion with drift, Electron. J. Probab. 19 (2014), no. 63, 27. MR 3238783, DOI 10.1214/EJP.v19-3169
R. K. Getoor, Excessive measures, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1093669, DOI 10.1007/978-1-4612-3470-8
R. K. Getoor and M. J. Sharpe, Naturality, standardness, and weak duality for Markov processes, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 1, 1–62. MR 756804, DOI 10.1007/BF00534082
Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. MR 1943877, DOI 10.1007/978-3-662-05265-5
Olav Kallenberg, Splitting at backward times in regenerative sets, Ann. Probab. 9 (1981), no. 5, 781–799. MR 628873
Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169, DOI 10.1007/978-1-4757-4015-8
H. Kaspi, On the symmetric Wiener-Hopf factorization for Markov additive processes, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 2, 179–196. MR 650610, DOI 10.1007/BF00531742
H. Kaspi, Excursions of Markov processes: an approach via Markov additive processes, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 2, 251–268. MR 714146, DOI 10.1007/BF01844609
Przemysław Klusik and Zbigniew Palmowski, A note on Wiener-Hopf factorization for Markov additive processes, J. Theoret. Probab. 27 (2014), no. 1, 202–219. MR 3174223, DOI 10.1007/s10959-012-0425-4
Victoria Knopova and René L. Schilling, A note on the existence of transition probability densities of Lévy processes, Forum Math. 25 (2013), no. 1, 125–149. MR 3010850, DOI 10.1515/form.2011.108
Alexey Kuznetsov, Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes, Ann. Appl. Probab. 20 (2010), no. 5, 1801–1830. MR 2724421, DOI 10.1214/09-AAP673
Andreas E. Kyprianou, Fluctuations of Lévy processes with applications, 2nd ed., Universitext, Springer, Heidelberg, 2014. Introductory lectures. MR 3155252, DOI 10.1007/978-3-642-37632-0
Andreas E. Kyprianou, Stable Lévy processes, self-similarity and the unit ball, ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 1, 617–690. MR 3808900, DOI 10.30757/alea.v15-25
Andreas E. Kyprianou, Deep factorisation of the stable process, Electron. J. Probab. 21 (2016), Paper No. 23, 28. MR 3485365, DOI 10.1214/16-EJP4506
A.E. Kyprianou, V. Rivero, and W. Satitkanitkul, Stable Lévy processes in a cone, arXiv:1804.08393, 2018
Lijun Bo and Chenggui Yuan, Stability in distribution of Markov-modulated stochastic differential delay equations with reflection, Stoch. Models 32 (2016), no. 3, 392–413. MR 3505450, DOI 10.1080/15326349.2016.1155463
Bernard Maisonneuve, Exit systems, Ann. Probability 3 (1975), no. 3, 399–411. MR 400417, DOI 10.1214/aop/1176996348
Sean P. Meyn and R. L. Tweedie, Stability of Markovian processes. II. Continuous-time processes and sampled chains, Adv. in Appl. Probab. 25 (1993), no. 3, 487–517. MR 1234294, DOI 10.2307/1427521
Masao Nagasawa, Time reversions of Markov processes, Nagoya Math. J. 24 (1964), 177–204. MR 169290, DOI 10.1017/S0027763000011405
P. Ney and E. Nummelin, Markov additive processes. I. Eigenvalue properties and limit theorems, Ann. Probab. 15 (1987), no. 2, 561–592. MR 885131
P. Ney and E. Nummelin, Markov additive processes II. Large deviations, Ann. Probab. 15 (1987), no. 2, 593–609. MR 885132
N. U. Prabhu, Stochastic storage processes, 2nd ed., Applications of Mathematics (New York), vol. 15, Springer-Verlag, New York, 1998. Queues, insurance risk, dams, and data communication. MR 1492990, DOI 10.1007/978-1-4612-1742-8
Víctor Rivero, Recurrent extensions of self-similar Markov processes and Cramér’s condition. II, Bernoulli 13 (2007), no. 4, 1053–1070. MR 2364226, DOI 10.3150/07-BEJ6082
L. C. G. Rogers, Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains, Ann. Appl. Probab. 4 (1994), no. 2, 390–413. MR 1272732
Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520
Michio Shimura, Excursions in a cone for two-dimensional Brownian motion, J. Math. Kyoto Univ. 25 (1985), no. 3, 433–443. MR 807490, DOI 10.1215/kjm/1250521064
Robin Stephenson, On the exponential functional of Markov additive processes, and applications to multi-type self-similar fragmentation processes and trees, ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 2, 1257–1292. MR 3867206, DOI 10.30757/alea.v15-47
John B. Walsh, Markov processes and their functionals in duality, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), 229–246. MR 329056, DOI 10.1007/BF00532535
Ward Whitt, Stochastic-process limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002. An introduction to stochastic-process limits and their application to queues. MR 1876437, DOI 10.1007/b97479
Ward Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5 (1980), no. 1, 67–85. MR 561155, DOI 10.1287/moor.5.1.67